The cartesian form of a complex number with high power index

Question

In the example I have : write $z^{2018}$ in its cartesian form .

$z= \frac{\sqrt 2}{2}(1-i)$

What are the steps that I should follow to solve such thing?

(HINT in the bottom of the page : remember that 2018 = 2016 + 2 = 8 ⋅ 252 + 2)

Answer 1

Hint: Use the hint, together with the fact that $\left(\frac{\sqrt2}2(1-i)\right)^8=1$.

Answer 2

If you're asked to compute $z^n$, always work in polar form. If you are asked to give the answer in Cartesian form, that's no problem; just convert at the end. We have $z=e^{-\pi i/4}$ (the modulus is $1$, but we'd still succeed even if it weren't). Thus $z^8=1$. That's the point of the hint: $z^{2018}=z^2=e^{-\pi i/2}=-i$.